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Transcript
6. Show that consecutive angles are supplementary What Makes a Quadrilateral a Parallelogram? What do you see? Are both pairs of opposite sides parallel? In This Picture… Is one pair of opposite sides congruent and parallel? Are both pairs of opposite sides congruent? Are both pairs of opposite angles congruent? What is this Picture? What is this? What is this? Is a Pentagon a Parallelogram? NO! GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 3. ANSWER Two pairs of opposite sides are equal. Therefore, the quadrilateral is a parallelogram. By theorem 8.7 GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 4. ANSWER By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram. GUIDED PRACTICE for Examples 2 and 3 5. For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning. SOLUTION 2x = 10 – 3x 5x = 10 x=2 By Theorem 8.6 [ Diagonals in bisect each other ] Add 3x to each side Divide each side by 5 6. Show that consecutive angles are supplementary Game Time: Name that Theorem Game Time: Name that Theorem Game Time: Name that Theorem Game Time: Name that Theorem Game Time: Name that Theorem 6. Show that consecutive angles are supplementary Game Time: Name that Theorem EXAMPLE 4 Use coordinate geometry Show that quadrilateral ABCD is a parallelogram. SOLUTION One way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9. First use the Distance Formula to show that AB and CD are congruent. AB = [2 – (–3)]2 + (5 – 3)2 = 29 CD = (5 – 0)2 + (2 – 0)2 29 = EXAMPLE 4 Use coordinate geometry Because AB = CD = 29 , AB CD. Then use the slope formula to show that AB CD. 5 – (3) 2 2 Slope of CD = 2 – 0 Slope of AB = 2 – (–3) = = 5–0 5 5 Because AB and CD have the same slope, they are parallel. ANSWER AB and CD are congruent and parallel. So, ABCD is a parallelogram by Theorem 8.9. EXAMPLE 4 GUIDED PRACTICE for Example 4 6. Refer to the Concept Summary. Explain how other methods can be used to show that quadrilateral ABCD in Example 4 is a parallelogram. SOLUTION Find the Slopes of all 4 sides and show that each opposite sides always have the same slope and, therefore, are parallel. Find the lengths of all 4 sides and show that the opposite sides are always the same length and, therefore, are congruent. Find the point of intersection of the diagonals and show the diagonals bisect each other. EXAMPLE 4 GUIDED PRACTICE K for Example 4 DK and TA are congruent and parallel. So, TDKA is a parallelogram by Theorem 8.9. A D T DK = [-4 – (0)]2 + (1 – 8)2 = 65 TA = [4 – (8)]2 + (-1 – 6)2 = 65 In Conclusion… Don’t forget your homework. • Pg 526 # 1-3, 11-14